Monotone penalty approximation of extremal solutions for quasilinear noncoercive variational inequalities

This paper is about a monotone approximation scheme for extremal (least or greatest) solutions of the following variational inequality:$\text{u∈K:〈Au+F(u),v−u〉⩾0,}\text{∀v∈K,}$in the interval between some appropriately defined sub- and supersolutions. The variational inequality is approximated by a sequence of penalty equations. The extremal solutions of the penalty equations, constructed iteratively and forming a monotone sequence, are proved to converge to the corresponding solutions of the original inequality. We note that no monotoneity assumption on the lower-order term F is imposed.